NASA Technical Memorandum !0_3181 .............

,-;-----_- A Viscoplastic Model With_Application to

___./:_ LiF-22 .____..... _....._C___a_F2 % Hypereutectic Salt _ ..............................................

A.D. Freed ........................

Lewis Research Center _.

Cleveland, Ohio .............

and

K.P. Walker

Engineering Science Software, Inc,

Smithfield, Rhode Island ....

Prepared for the

3rd International Conference on Constituti_ve Laws for_ Engineering Materials:

Theory and Applications and Workshop on Innovative Use-of Materials in

Industrial and Infrastructure Design and Manufacturing

, sponsored by the University of Arizona

Tucson, Arizona, January 7-12, 1991

_UOE:L WITH

-(_!A_A-Tm-IO-_IgI) A VTSCOPLASTIC " -

APPLICATION T_7 kiF-?2 p_RCENT C,_E?

i-iYpFRF-UTECTIC SALT (NASA) i3 p C_CL 20K

N90-23 770 ......... _':'

Llnc1 ,_s

63/39 0266267 .............

....................... m ................... # _, = 3_

A Viscoplastic Model with Application to LiF-22%CaF2

Hypereutectic Salt

A. D. Freed

NASA Lewis Research Center

Cleveland, Ohio

K. P. Walker

Engineering Science Software, Inc.

Smithfield, Rhode Island

Abstract

A viscoplastic model for class M (metal-like

behavior) materials is presented. One novel

feature of this model is its use of internal variables to change the stress exponent of creep

(where n _ 5) to that of'natural' creep (where

n = 3), in accordance with experimental observations. Another feature is the introduction of a coupling in the evolution equations

of the kinematic and isotropic internal variables, making thermal recovery of the kinematic variable implicit. These features enable

our viscoplastic model to reduce to that of

steady-state creep in closed form. In addition,

the hardening parameters associated with the

two internal state variables (one scalar-valued,

the other tensor-valued) are considered to be

functions of state, instead of being taken as

constant-valued. This feature enables each internal variable to represent a much wider spectrum of internal states for the material. We

apply our model to a LiF-22%CaF2 hypereutectic salt, which is being considered as a thermal energy storage material for space-based

solar dynamic power systems.

1 Introduction

The primary power supply for Space Station

Freedom will be photovoltaic panels (75 kW

total--phase I implementation). Under consideration as an option, supplemental power

will be supplied by reflecting the Sun's energy

into two solar dynamic power modules (25 kW

each--phase II implementation) [1]. Within

the solar receiver of each power module will be

a network of working fluid tubes. This working fluid will transport thermal energy to heat

engines for electric power generation. Encasing each working fluid tube will be a string

of canisters. The basic function of these canisters is to store heat when they are exposed

to sunlight, and then release this heat when

the vehicle is in eclipse. Typically, this energy storage-release cycle is attained through

a thaw-freeze cycle of a special material contained within the canisters.

The attractive thermophysical properties of

fluoride salts make them potential candidates

for use as thermal energy storage materials in

space-based solar dynamic systems [2]. Currently, the LiF-21mol%CaF_ eutectic mixture

is being considered for this purpose. A major

problem that arises with the use of fluoride

salts is the very large volume expansions that
theyundergouponmelting(~ 30%);plus,the

fairly large volume expansions that they incur

upon heating from room temperature to melting (,_ 5%). These volume expansions can lead

to a stress buildup on the walls of a canister,

especially when pockets of molten salt are entrapped between the containment vessel and

unmelted salt. As a result, localized distortion or rupture of a canister could occur if the

unmelted salt does not deform easily to accomodate this volume change.

This paper presents the development of a

viscoplastic model for a LiF-22mol%CaF2 hypereutectic salt based upon the experimental

data of RAJ & WHITTENBERGER [3]. This viscoplastic model, and a similar one for the containment vessel (IIaynes Alloy 188, a class A--

alloy behavior, i.e. dynamic strain aging--

material), will be used in future stress analyses

of a canister to address the possiblity of localized distortion or rupture; thereby, assessing

the durability of these canisters.

2 Stress-Strain Relations

Stress (rij is assumed to be related to infinitesimal strain eij through the isotropic constitutive equations

v = 0 (1)

aii= 3x(eii - a AT 6ii) (2)

with shear p and bulk t¢ elastic moduli, where

S_j = _rij - o'_k6ij/3

eij = eij - ekk6i.i /3

denote the deviatoric stress and strain. Equation 1 charactcrizes the deviatoric stress response, while Eqn. 2 characterizes the hydrostatic stress response. The inelastic ePj and

thermal a AT 6ij strains are eigenstrains that

represent deviations from deviatoric and hydrostatic elastic behaviors, respectively. The

constant a is the mean coefficient of thermal

expansion, while AT -- T - To represents a

difference between the current temperature T

and some reference temperature To. Herein, T

represents an absolute temperature in degrees

Kelvin. The quantity gij is the Kronecker

delta. Repeated Latin indices are summed

over from 1 to 3 in the usual manner.

Young's modulus E and Poisson's ratio v

are often the elastic moduli that are experimentally determined. The expressions

E

p --

2(1 + v)

E

K --

3(1 - 2v)

define their interdependence, since only two

elastic moduli are independent for isotropic

materials.

3 Viscoplastic Theory

The mathematical structure for the theory

of viscoplasticity considered in this paper

is a special case of a much more general

structure that was discussed by FREED

CItABOCHE [4]. The theory considered admits two internal state variables; they are:

i) the (scalar-valued) yield strength Y >_ 0,

and i 0 the (deviatoric tensor-valued) back

stress Bij. The yield strength accounts for

isotropic hardening effects, while the back

stress accounts for kinematic (flow-induced

anisotropic) hardening effects. These internal

variables are considered to evolve phenomenologically through competitive processes associated with strain hardening, strain-induced

dynamic recovery, and time-induced thermal

recovery. The choice of a yield strength (over

that of a drag strength) for the isotropic variable is consistent with results from an earlier

study done by the authors [5].
The flow equation for inelastic strain is given

by

_ij =

with the effective stress

Eij= Sij - Bij

establishing the direction of inelastic strain

rate. A dot placed over a variable denotes its

time rate-of-change.

The norms (or magnitudes) of this theory

are defined as

where hj is any deviatoric strain-like tensor,

viz. _iij, and where Jij is any deviatoric stresslike tensor, viz. Sij, Bij and Eij. These norms

are of the yon Mises type, where the coefficients are chosen to scale the theory for tension.

The evolution equations characterizing the

internal state of the material are given by

and

? = h(ll_Pll - dtl_Pll Y - r) (5)

where H > 0 and h > 0 are the strain hardening parameters, D > 0 and d >__0 are the

dynamic recovery parameters, and R > 0 and

r > 0 are the thermal recovery parameters. No

parameter can be negative-valued and satisfy

the thermodynamics, in general.

Defining functional forms for the six parameters H, h, D, d, R and v, along with a kinetic equation for IlgPl[, results in a specific

viscoplastic model. Such a model is presented

in the next section for class M materials.

4 Viscoplastic Model

Material functions that characterize a viscoplastic model for class M materials are given

below. A discussion of these material functions is presented in the following subsection.

Derivations of the material functions, where

noted, are presented in the Appendix.

A Zener type kinetic equation for the evolution of inelastic strain is considered, i.e.

II_Pll= 0z (6)

where d(T) > 0 acts as a thermal diffusivity, and Z(Sij, Bij, Y) >__0 is referred to as the

Zener parameter. 1 The ZENErt-HoLLOMON [6]

hypothesis assumes that the kinetic equation

can be represented as a product of two functions t9 and Z; the first is dependent only

on temperature, while the latter is dependent

only on stress and the internal state.

For the thermal diffusivity, we will use the

relationship

Tt<_T<T,.

o = (7)

O<T<_Tt

which was derived by MILLER [7]. Here k is the

universal gas constant (k = 8.314 J/mol-K), Q

is the activation energy, Tm is the absolute

melting temperature, and Tt is the absolute

transition temperature. For the Zener parameter, we shall consider the relationship

Z : A sinh3 ((IIE_ - Y)) (8)

1 The preference of a temperature-dependent activation energy over a stress-dependent activation energy

is discussed in a companion paper by Fal_EI), RaJ &

WALKEa, which is also published in these conference

proceedings.
with frequency coefficient A and drag strength

K. The Maeauley bracket operator ([[_[[- Y)

has either a value of 0 whenever [[Z[[ _< Y

(defining the elastic domain), or a value of

[[E[[- Y whenever [[Z[[ > Y (defining the viscoplastic domain), with [[E[[ = Y establishing

the yield surface.

The three material parameters associated

with the evolution of back stress for this particular model are taken to be

H = (0.01+0.99¢m)E

D = ____ y

bY

R = 0

(9)

where m and y are positive-valued material

constants, and where b and _ are functions of

state defined in Eqns. 11 and 12. The function for dynamic recovery D is derived in the

Appendix. Since R = 0, there is no thermal recovery term in the evolution equation for back

stress in this model.

The three material parameters associated

with the evolution of yield strength for this

particular model are taken to be

h = clsinh3-n(y-_)

b_

d-- c2-- (10)

Y

r = A0sinhn(_)

where C, cl, c2 and n are positive-valued material constants. The function for thermal recovery r is derived in the Appendix. It is worth

noting that unlike the recovery parameters,

the functional form of a hardening parameter

cannot be determined a priori, but must be

postulated based upon experimental observations.

The fraction of applied stress which is attributed to the 'back stress' under steady-state

B

Surface

Figure 1: Geometric representation of state

space.

conditions is determined to be

b = 1 YK"-its (_'-_)] (11) - y- T slnh inh n/3 Y

where the material constant y represents the

percentage of applied stress which is attributed to the 'yield strength' under steadystate conditions. The derivation of Eqn. 11 is

given in the Appendix.

A measure for the distance between the current point Bij in stress space and its imaging

point :_I_/DIIEII on the bounding surface is

given by

(12)

where 0 < _ < 1. This relationship comes from

considerations of state-space geometry for our

model, as illustrated in Fig. 1. The cord between the current state of back stress and its
imagepoint of length2_/D is in the direction that the back stress evolves, which is the

viscoplastic analog to the kinematic evolution

in a two-surface MRoz [8] model for plasticity. A value of _ = 0 implies that Bij is on the

bounding surface, and that Sij is either on this

surface or outside of it. A value of _ = 1 also

implies that Bij is on the bounding surface,

but now Sij is inside this surface, i.e. unloading from a saturation state has just begun.

4.1 Discussion of the Model

In total, there are three temperaturedependent material constants pertaining to

thermoelasticity, viz. a, _ and #, ten

temperature-independent material constants

pertaining to viscoplasticity, viz. A, C, Cl, c2,

K, m, n, Q, Tt and y, and one initial condition, viz. Yo. Of these ten viscoplastic constants, five are determined from steady-state

creep data, viz. A, C, n, Q and Tt, while

the remaining five must be determined from

data involving transient behavior. By definition, Y0 = 0 in an annealled state. The material constants for LiF-22%CaF2 used in the

analyses presented in this paper are listed in

Table 1. Data for the elastic constants [9] presented therein were correlated over the temperature range of 300 to 950 K, with melting

occurring at 1042 K. There is danger in extrapolating their values outside of this range,

especially for u whose expression is in direct

violation with physics for T < 150 K.

An important consequence of the viscoplastic model presented above is that the Zener

parameter defined in Eqn. 8 reduces in closed

form to the Zener parameter associated with

GAROFALO'S [10] creep equation under steady

state conditions, i.e.

where the subscript 'ss' denotes steady state,

Constant Units

O_

MPa

12

A

C

106 K-1

S-1

MPa

Value

18 + 0.03T

52, 000 - 29T

0.65 - 0.001T

5 x 1015

27

cl MPa 50,000

C2 -- 0

K MPa 8

m

TI

Q J/mol.

K

K

T_

Tt

y m

Y0 MPa

320,000

1042

85O

0.1

0

Table 1: Constants for LiF-22%CaF2.

and C is the power-law breakdown stress. The

capability of Zener's kinetic equation (6) to

correlate experimental creep data using the

relationships for t9 and Z,, given in Eqns. 7

and 13 is demonstrated in Fig. 2.

The hardening parameter H given in Eqn. 9

is similar in form and function to the tangent

modulus of the DAFALIAS-PoPov [11] twosurface theory of plasticity. Both allow for a

smooth transition between elastic and inelastic behavior. The exponent m characterizes

the strength of curvature in a typical hysteresis loop. Since there are no cyclic data available for LiF-22%CaF2, an assumed value for

m is given in Table 1.

As previously mentioned, there is no 'explicit' representation for the thermal recovery

of back stress in our model. But take note

that the surface which bounds the state of

back stress--characterized by D--is a function of the yield strength in Eqn. 9. Consequently, since the yield strength thermally re-
F==¢

N

1029

102s

10 z7

102_

1025

102,*

10 23

1022

1021

102o

1019

10ts

1017

1016

1015

1014

1013 _

10t2

1011

101o

10 o

m

_ +500K

- _600K

- x700K

o 800K

[] 900K

_ o 950 K

- A 1015 K

10 _ 10 2 10 3

Oss (MPa)

Figure 2: Steady-state kinetic response of

LiF-22%CaF2. Data are from RAJ _c WHITTENSERCER[3].

covers, thermal recovery of the back stress is

'implicit'. Unfortunately, inelastic strain must

evolve in order for this implicit recovery to

take place. Even though this is not a physically accurate representation of thermal recovery, it is a very useful approximation to it. It

is useful because it allows the effect of thermal

recovery to be modeled, while still enabling

our viscoplastic model to reduce to a steadystate creep model (in particular, Garofalo's)

in closed form. This enables us to use creep

data directly for the purpose of characterizing

about half of our model's material constants--

a huge benefit.

In the domain of power-law creep, the hardening and thermal recovery parameters, h and

r, given in Eqn. 10 are similar in form and

function to the hardening/recovery relationships proposed and experimentally verified by

MITRA _ MCLEAN [12]. Here the stress dependence of recovery, as defined by the product h r, has an exponent of 3. This is in accordance with the 'natural' theory of creep

for thermally-assisted dislocation climb [13].

Our particular hardening and thermal recovery parameters extrapolate the Mitra-McLean

results into the domain of exponential behavior. The viscoplastic model of ROBINSON [14]

also employs the hardening/recovery concept

of Mitra and McLean, but in the evolution

equation for back stress.

In the viscoplastic model of MILLER [7], the

dynamic recovery parameter d was introduced

to provide a coupling between the back stress

and drag strength so that differences between

the monotonic and cyclic stress-strain curves

could be accounted for. This distinction between monotonic and cyclic behavior is a phenomenon that CHABOCHE el al. [15] refer to

as the strain memory effect, but one that they

model in a completely different manner than

the approach used by Miller. Our particular

function for d given in Eqn. 10 differs from

Miller's in form, but not in purpose. In both of
thesemodels,d_s = 0 at steady state, thereby

enabling the viscoplastic models to reduce to

well-known creep models in closed form. In

our model, _ (not HBID is the parameter chosen to account for the oscillatory changes in

of the internal state of the material during

cyclic loadings. The ratio b/y proportions this

coupling according to the relative strength capability of the back stress to yield strength.

Because there are no cyclic data available for

LiF-22%CaF2, this coupling effect is not considered in this paper.

The material constant y establishes, among

other things, the capability of a material to

cyclically harden. The greater the value of

y, the greater is this capability. An assumed value for y is given in Table 1 for

LiF-22%CaF2 because of a lack of cyclic data.

The expression for b in Eqn. 11 (defining

that portion of the applied stress which is attributed to the back stress under conditi,_,s

of steady state) is a direct consequence of thc

following hypothesis: it is the internal state

of stress that accounts for the difference b_-

tween 'natural' (with a stress exponent of 3,

cf Eqn. 8) and observed (where n _ 5, cf.

Eqn. 13) power-law creep behavior in class M

materials. This internal stress results from

a honeycomb-like dislocation structure composed of hard cell walls of high dislocation density which surround soft cell interiors of relatively low dislocation density [16]. The 'natural' theory of creep assumes a homogeneous

dislocation network. Notice that when n = 3

(as is the case for class A materials, which typically have no cellular dislocation structure),

the fraction of back stress to applied stress at

steady state becomes constant-valued; in particular,

K

b= l-y--_

Otherwise, b varies with material strength, for

which the yield strength Y is an appropriate

measure. It is this feature of the variable b that

100

75

5O

25

o Data Theory I

0

450 650 850 1050

T (K)

Figure 3: Yield strength (0.2% offset) of

LiF-22%CaF2. t = 2 x 10-4 s -1. Data are

from RAJ _ WHITTENBERGER. [3].

distinguishes this viscoplastic model from the

previous ones developed by the authors [5, 17].

5 Applications

There are very few experimental data for

LiF-22%CaF2 that one can use to validate a

model such as the-one given above. About

the only data that are available, besides those

which appear in Fig. 2, are yield strength data.

Figure 3 presents the predictive capability of

our model, as it pertains to the yield strength.

The yield strength ay at 500 K, and the fact

that about 15% strain was required to attain
t:)

120

8O

40

0

' I ' I ' i ' I ' I

, I , l, I, I .1

1.0

-40 ,

-1.5-1.0-0.5 0.0 0.5 1.5x10 -3

£- ¢t AT

Figure 4: Out-of-phase TMF response predicted for LiF-22%CaF2. Tmi, = 400 K.

Tmax = 800 K. tmech ----- _--o_T = lxl0 -4 s -1.

steady-state in this test, were facts used to set

values for the material constants K and cl, respectively. The model predictions for all other

temperatures are just that, predictions. The

agreement between theory and experiment is

quite good.

To demonstrate some of the capabilities

of our model, the response to the first few

cycles of a typical thermomechanical fatigue

(TMF) loading history are presented in Fig. 4.

Unfortunately, there are no experimental data

to substantiate this prediction one way or the

other. Nevertheless, the predicted shape of

this hysteresis loop is consistent with shapes

observed from similar experiments done on

metals [18]. In short, our model is capable of

representing the complexities of nonisothermal

material behavior while using temperatureindependent viscoplastic material constants.

6 Conclusions

A viscoplastic model has been proposed for

class M materials. This model has been applied to a LiF-22%CaF2 hypereutectic salt.

Features of this model include:

s Closed form reduction to creep theory under steady-state conditions.

• Internal state variables account for the

difference in the stress dependence of 'natural' and observed creep behavior.

• Two internal state variables are employed.

1. The yield strength accounts for

isotropic hardening and softening effects.

2. The back str_ accounts for kinematic hardening and softening effects.

• These internal variablesevolve according to the competitive processesof strain

hardening, and dynamic and thermal recovery.

• A Mitra-McLean hardening/recovery format is used in the evolution of yield

strength.

• DynamO(recovery of the yieid strength

accounts for monotonic/cyclic interaction

(or strain-memory) effects.

• Thermal recovery of the back stress is

taken to be implicit; thereby, simplifying

the structure and characterization procedure of the model.
• Thestrainhardeningparameterfor back

stressis comparablewith the tangent

modulusin the Dafalias-Popovtwosurfacetheoryofplasticity.

References

[1] M O DUSTIN. J M SAVlNO, D E LACY, R P MIGRA, A L JUHASZ AND (_ E COLES, "Advanced

Solar Dynamic Space Power Systems Perspectives, Requirements, and Technology Needs," Solar Engineering-1987, eds D Y Goswami, K

Watanabe and H M Healey, ASME, New York,

1987, pp 574-588.

[2] A K MISRA, "Fluoride Salts as Phase Change

Materials for Thermal Energy Storage in the

Temperature Range 1000-1400 K: Thermal

Analysis and Heat of Fusion Measurements," J

Electrochem Soc, Vol 135, 1988, pp 850-854.

[3] S V RAJ AND J D WHITTENBERGER, "Deformation of As-cast LiF-22mol°ACaF2 Hypereutectlc

Salt Between 500 and 1015 K," to appear in:

Mater Sci Enfl, 1990.

[4] A D FREED AND J-L CtlAnOCHE, "Viscoplastlclty: A Thermodynamic Formulation," NASA

TM-I 0_388, 1989.

[5] A D FREED AND IS: P 1A'ALKER, "Steady-State

and Transient Zener Parameters in Viscoplasticlty: Drag Strength Versus Yield Strength,"

NASA TM-10_._87, 1990. To appear in: Appl

Mech Rev, 1990.

[6] C ZENER AND J H HOLLOMON, "Effect of Strain

Rate Upon Plastic Flow of Steel," J Appl Phys,

Vol 15, 1944, pp 22-32.

[7] A MILLER, "An Inelastic Constitutive Model for

Monotonic, Cyclic, and Creep Deformation: Part

I--Equations Development and Analytical Procedures," J Eng Mater Teehnol, Vol 98, 1976,

pp 97-105.

[8] Z MROZ, "On the Description of Anisotropic

Workhardening," J Mech Phys Solids, Vol 15,

1967, pp 163-175.

[9] A WOLFENDEN, S V l:_AJ. G LASTRAPES AND

M B DUGGAN, "Temperature Dependence of the

Elastic Moduli and Damping for Polycrystalline

LIF-22%CaF2 Eutectic Salt," submitted to: J

Mater Sei.

[10] F GAROrAI.O, "An Empirical Relation Defining

the Stress Dependence of Minimum Creep Rate

in Metals," Trans Metall Soc AIME, Vol 227,

1963, pp 351-356.

[11] Y F DAFXLIA$ AND E e PoPov, "Plastic Internal

Variables Formalism of Cyclic Plasticity," J Appl

Mech, Vol 43, 1976, pp 645-651.

[12] S K MITRA AND D McLEAN, "Work Hardening

and Recovery in Creep," Proc Royal Soc, London, Ser A, Vol 295, 1966, pp 288-299.

[13] J WEERTMAN, "High Temperature Creep Produced by Dislocation Motion," Rate Processes

in Plastic Deformation o] Materials, eds J C M

Li and A K Mukherjee, ASM, Cleveland, 1975,

pp 315-336.

[14] D N ROBINSON, "A Unified Creep-Plasticity

Model for Structural Metals at High Temperatures," ORNL TM-5969, 1978.

[15] J-L (._ItABOCHE, K DANG-VAN AND G (,._ORDIER,

"Modelization of the Strain Memory Effect on

the Cyclic ltardening of 316 Stainless Steel,"

SMIRT-5, Div L, Berlin, 1979.

[16] W D _NIx AND B ILSCHNER, "Mechanisms Controlling Creep of Single Phase Metals and Alloys," Strength of Metals and Alloys, Vol 3, eds

P tiaasen, V Gerold and G Kostorz, Pergamon

Press, New York, 1980, pp 1503-1530.

[17] A D FREED AND K P _'ALKF_R, "Refinements

in a Viscoplastic Model," Viscoplastic Behavior

of New Materials, eds D Hui and T J Kozik,

ASME, New York, PVP-Vol 184, MD-Vol 17,

1989, pp 1-9.

[18] V (.] RAMASVeAMY, R H VAN STONE, L T DAME

AND J H LAFLEN, "Constitutive Modeling for

Isotropic Materials," NA SA CR 175004, 1985.

[19] J J BLASS AND W N FINDLEY, "Short-time, Biaxial Creep of an Aluminum Alloy with Abrupt

Changes of Temperature and State of Stress," J

Appl Mech, Vol 38, 1971, pp 489-501.

[20] J (_ADEK, "The Back Stress Concept in Power

Law Creep of Metals: A Review," Mater Sci Eng,

Vol 94, 1987, pp 79-92.

[21] T C LOWE AND A K MmLER, "The Nature of

Directional Strain Softening," Scr Metall, Vol 17,

1983, pp 1177-1182.
Appendix

Much of the development of our viscoplastic

model is predicated upon the assumption that

IIBII. = bllSll and Y. = YlISII (14)

where y is taken to be a constant, but not b. It

also makes use of the fact that the back stress

Bij is coaxial with the deviatoric stress Sij

under steady-state conditions (a direct consequence of Eqns. 3 and 4, and in agreement with

experimental evidence [19]), and therefore one

can write

I1 11-- IIS- Bib, = IISII- IIBII. (15)

which is not valid, in general.

By definition, Bii = 0 at steady state, and

since R = 0 Eqn. 4 reduces to

1

D,, = IIBIt,,

Combining this result with Eqn. 14 leads

directly to the dynamic recovery parameter

given in Eqn. 9. Because D is a strength measure (it establishes the magnitude of the limiting state of back stress), it is appropriate that

it depend on the yield strength and not on the

magnitude of back stress.

Likewise, 1} = 0 at steady-state by definition, and therefore

,',, = I1 "11-= o z.

where we make use of the self-imposed constraint that dj8 - 0 (satisfied by the fact that

¢,, = 0). Combining Eqns. 13 and 14 with this

result leads to the thermal recovery parameter

given in Eqn. 10.

Equating the transient Zener parameter of

Eqn. 8 with its steady-state counterpart given

in Eqn. 13 under steady-state conditions, and

using Eqns. 14 and 15, one obtains the relationship

b + y = 1 - [_,,sinh -1 [sinh '*/3

(16)

which when written in terms of Y,, via Eqn. 14

results in Eqn. 11 for b. Here sinh -1 is the

arcsinh, whose equivalent expression

sinh-t(x) = In (z +

is a useful representation when writing FORTRAN code.

A graphic representation of Eqn. 16 is presented in Fig. 5. In the domain of power-law

creep, Eqn. 16 reduces to a relationship that

(_ADEK [20] proposed and experimentally verified, where b and y were not distinguished one

from the other. In the domain of exponential

creep, Eqn. 16 implies that the fraction of internal stress (b + y)[[S[] to applied stress [[SI[

at steady state is nearly a constant. The ratio

K/C establishes this plateau, which suggests

an upper bound for y. As a rule of thumb, a

ratio of K/C _ 1/3 leads to an internal stress

that is between 40 and 50% of the applied

st_s_t steady state, which-is c0nsistent with

experimental observations [21]. In this study,

the value of K was determined by correlating

yield strength data, and was found to agree

with this rule of thumb.

10
1.00

0.75

+ 0.50

0.25

........ I ........ I ........

- K=8 :

(_plb (_'ult

........ [ ........ [ ........

0._0o_ 101 10 2 lO s

Os_ (MPa)

Figure 5: Response of Eqn. 16, where values

for the power-law breakdown stress _rplb ------C,

the ultimate strength (rult and the creep exponent n pertain to LiF-22%CaF2. Plots for

two different values of K are displayed.

11
National Aeronaulics and Report Documentation Page

Space Adminislration

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TM-I03181

5. Report Date 4. Title and Subtitle

A Viscoplastic Model With Application to LiF-22%CaF 2

Hypereutectic Salt

7. Author(s)

A.D. Freed and K.P. Walker

9. Performing Organization Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546-0001

6. Performing Organization Code

8. Performing Organization Report No,

E-5559

10. Work Unit No.

474-52-10

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

Prepared for the 3rd International Conference on Constitutive Laws for Engineering Materials: Theory and

Applications, and Workshop on Innovative Use of Materials in Industrial and Infrastructure Design and

Manufacturing sponsored by the University of Arizona, Tucson, Arizona, January 7-12, 1991. A.D. Freed,

NASA Lewis Research Center. K.P. Walker, Engineering Science Software, Inc., Smithfield, Rhode Island.

16, Abstract

A viscoplastic model for class M (metal-like behavior) matearials is presented. One :novel feature of this model is

its use of internal variables to change the stress exponent of creep (where n _5) to that of 'natural' creep (where

n = 3), in accordance with experimentaql observations. Another feature is the introduction of a coupling in the

evolution equations of the kinematic and isotropic internal variables, making thermal recovery of the kinematic

variable implicit. These features enable our viscoplastic model to reduce to that of steady-state creep in closed

form. In addition, the hardening parameters associated with the two internal state variables (one scalar-valued, the

other tensor-valued) are considered to be functions of state, instead of being taken as constant-valued. This

feature enables each internal variable to represent a much wider spectrum of internal states for the material. We

apply our model to a LiF-22%CaF 2 hypereutectic salt, which is being considered as a thermal energy storage

material for space-based solar dynamic power systems.

L

"17. Key Words (Suggested by Author(s))

Viscoplasticity

Creep

Salt

18. Distribution Statement

Unclassified - Unlimited

Subject Category 39

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21, No. of pages 22. Price*

Unclassi fled Unclassified A

NASAFORM1626OCT_ *For sale by the National Technical Information Service, Springfield, Virginia 22161